1. Let Y1, . .., Yn ~ f(y; 0). For the following densities f, find: . a sufficient statistic for 0; . the moment estimator of 0; . the MLE of 0; . the Fisher information I(0) = -nE 12 dzo log f (Y; 0) ); . the Cramer-Rao lower bound for an unbiased estimator of 0. (a) Poisson: f(y; 0) = 0 expi -01 for y E {0, 1, 2, ... } and 0 > 0. (b) Bernoulli: f(y;0) = 09(1 - 0)1-y for y E {0, 1} and 0 E (0, 1). (c) Geometric: f(y; 0) = 0(1 - 0)y-1 for y E {1, 2, ...} and 0 E (0, 1). (d) Exponential: f(y;0) = _exp {-} for y > 0 and 0 > 0